It's a tribute to Nicholas Hytner's mighty reign at the National that the news of his standing down outshone, or outdarkened, any opening last week. I would back Stephen Daldry. He has a wide range, a big fund of ideas and tremendous directorial flair. What's more, he's a real showman who lifts the spirits; he could get anyone to do anything; he would raise money. I'd also be happy to see the the entrepreneurial pizazz of Tom Morris combined with the directorial brilliance of Marianne Elliott in a dual directorship (reprising their Warhorse partnership). And I wouldn't rule out – even if he says he does – the talent of Dominic Cooke, whose last commissioned play for the Royal Court has just opened.

The plays of Anthony Neilson have run as a dark, dissenting thread through the repertoire of Cooke's Court. They've offered fractured stories (The Wonderful World of Dissocia), gruesome deaths (Relocated), unexplained switches of place and time, great grim glimmers, wilful obfuscations (always), a practice of devising during rehearsal (Neilson is a directing dramatist) and of leaving work (including titles) open till the last moment. His plays come closer than most to an under-advertised aspect of Cooke's Court project: an interest in breaking free of social realism and exploring the experimental tradition of Ionesco and Beckett.

Narrative is Neilson incarnate: exhilarating and irritating. The title is a signal that there will be no completed or conventional narrative. A rather brilliant disquisition on the Lascaux cave paintings proposes that a drawing of a man spearing a bison is only a story if it ends in death, and raises the interesting question of whether our hero is the beast or the human. On this are appended, along with fetching bison videos, episodes from broken tales. A couple are in mid-breakup, a girl inexplicably stabs her best friend. An actor may be on the brink of stardom as Elastic Man, a figure who poses, he explains, a greater imaginative challenge than Macbeth, since everyone knows what it's like to be a bit ambitious, but few have stretched their limbs to infinite lengths. Scenes swim in and out of inner landscapes and fantasy; different conversations overlap; movie and radio scripts take over characters' lives; every now and then someone sprouts horns. There is a fine comic riff on the anus, and a neat squib about North Korea.

The less-than-two-hour evening is delivered with bite and stylishness, greatly helped by Chahine Yavroyan's powerful lighting which floods Garance Marneur's sleek design of long, white surfaces and dark shards of glass. Yet the individually sharp scenes accrete without growing; all that increases is a sense of a hermetically sealed drama. Neilson might argue that frustrating our sense of an ending is the heart of his play, but he beats us over the head with it. With murderous panache.

The Royal Exchange is being bold. Rory Mullarkey is 25, the youngest dramatist to have work produced on the theatre's main stage, and his play Cannibals has set its sights on big beasts. It springs from a glimpse of a terrible era. In Moscow's modern history museum, Mullarkey saw a photograph from the 1930s in which a group of villagers bore the caption "Cannibals near Perm". The play offers a brief history of the former Soviet Union, a tale of 20th-century famine and war and predators, and a story of females struggling for power. It moves from eastern Europe to Manchester and offers both literal and metaphorical accounts of its title. Within minutes of opening, the play has delivered two murderous shocks.

The theatre was right to take a punt on Mullarkey. He has talent, for seeking out a real subject and for translating it into action on stage. His background as a specialist and translator in Russian and Ukrainian shines through in the authentic tough texture of individual exchanges. Yet in Cannibals he has – um – bitten off more than he can yet chew. The main disappointment in an over-episodic play is that the characters are so little surprising, so off-the-pegski. There is one holy fool, one icon painter, one crone in a headscarf.

Still, Michael Longhurst has given Cannibals a tremendous production, proving, after his great success with Constellations, what an enabling director he is. Chloe Lamford's massive design cannons through the height of the Royal Exchange. On to the stage rain carrots and spuds; body bits dangle down in bloodied plastic bags; flames shoot up the chains that hang from the roof; our heroine swings in a harness like another bit of meat on a butcher's hook. What an all-over pelting theatre in the round can provide.

John Tiffany doesn't like to repeat himself. First he directed the explosive Black Watch, later the concentrated intensity of Be Near Me and the rough and tumble reportage of Enquirer. Now he has seduced Broadway with wispy, wistful Once. Every now and then Tiffany makes you think you're watching an entirely new way of putting a musical together.

This staging of John Carney's film, with music and lyrics by Glen Hansard and Markéta Irglová and an expanded script by Enda Walsh, has two strong features. It's not so much event as music-driven. You hear rather than see characters growing close to each in the show's daintily arranged and best-known song, Falling Slowly. When an improvised band (which includes a barman and a banker) come together to make a recording, it moves gradually from tentativeness to confidence, guitar and piano, through strings and percussion, to make a lovely demonstration of co-operation and changing mood. The instrument-toting actors are throughout terrific.

All together now: Declan Bennett and Zrinka Cvitešić in Once at the Phoenix theatre. Photograph: Tristram Kenton Tristram Kenton/ Tristram Kenton

Once is also a feat of visual inclusiveness. Before the show, the audience is cajoled on to a stage thrumming with stamping and singing and violins, accordions and guitars. When the spectators are safely back in their seats, they see Bob Crowley's Irish pub set in its peat-coloured glory. Unusually exact but unfussy for a musical design, it's fuggy but spacious, with wood-covered walls hung with rows of black-and-white photographs. The success of the design is not Crowley's alone. The lighting by Natasha Katz is marvellous, expanding and contracting the focus, creating pools of intimacy or isolation; sinking the mood; conjuring up a mini night-time Dublin in blue fairylights; adding to the credibility of that pub with wall lamps that shed a yellow glow.

So far so appealing. But the core of the show is more feeble. The music is not interesting enough to swell the evening. The first number is ill-advisedly sung immediately after a beautiful plain rendering of Patrick Kavanagh's On Raglan Road: it leaves you wondering whether the young guitarist is actually supposed to be any good. The story is shaky. A down-in-the-mouth Dublin guitar player who mends Hoovers for a living is befriended by an ultra-pure Czech pianist who heals his heart and soul and leaves him. The doleful wispiness of this matters not a jot. It's a kind of boldness. The whimsy with which it is veined does matter. He's called Guy; she's called Girl. Before she plays anything, she says hello to her piano. Tiffany's skill scatters gold dust over the weaknesses.

This article was amended on 15 April. Once is set in Dublin not Belfast as originally stated.